I can't remember now which type of modem you have & I can't get access to your win audit report easily as I'm currently using my laptop - if you have a modem with a network port on the back (as well as USB) & you also have a network card (I'm fairly sure you do), then you should be able to connect the modem to the network card using a standard CAT5 network cable, instead of USB. This will then allow you to have Internet access in safe mode as no special drivers are required for the modem this way.
I'm fairly certain if you do have a network card, then it's an integrated one i.e part of your motherboard. This means the port will be on the rear I/O panel, probably near the USB ports. The port will look like a phone jack but slightly bigger.
If your system does prove to be stable in safe mode, then this would suggest that the problem is with either a driver or a program as opposed to a hardware problem. However, I would advise you run it in safe mode a few days before drawing that conclusion. Obviously this is going to be a more viable option if you can get Internet access in safe mode.
As for a new computer - you can certainly buy a new computer for not much more money. However, a lot of the time you won't get a decent power supply. Also, the motherboards tend to be ones with a lot of integrated components (eg video/sound/network etc). This often means that there are very few PCI or PCI Express ports on the board which will limit the potential to upgrade or add other cards in the future. (In particular, video).
You may also find that it comes with generic memory. The actual difference in price between generic memory & decent quality (eg Kingston/Crucial/Corsair/OCZ) memory a just a few quid. But decent memory is likely to be more reliable that generic & most importantly, comes with a lifetime warranty. (Generic stuff does not).
You will also find that a lot of system manufacturers do not tell you what make/model of motherboard is in the system (assuming they are using a 3rd party board to start with).
If you decide to go out & buy a new complete system, then I would advise you try Dell - Dell tend to be very reasonably priced, plus you can pretty much select all the components individually i.e custom build it to a large extent.